MathDB

Robot "Mag-o-matic" manipulates

101

glasses, displaced in a row whose positions are numbered from

1

101

. In each glass you can find a ball or not. Mag-o-matic only accepts elementary instructions of the form

(a;b,c)

, which it interprets as "consider the glass in position

a

: if it contains a ball, then switch the glasses in positions

b

and

c

(together with their own content), otherwise move on to the following instruction" (it means that

a,\,b,\,c

are integers between

1

and

101

, with

b

and

c

different from each other but not necessarily different from

a

). A \emph{programme} is a finite sequence of elementary instructions, assigned at the beginning, that Mag-o-matic does one by one. A subset

S\subseteq \{0,\,1,\,2,\dots,\,101\}

is said to be \emph{identifiable} if there exists a programme which, starting from any initial configuration, produces a final configuration in which the glass in position

1

contains a ball if and only if the number of glasses containing a ball is an element of

S

. (a) Prove that the subset

S

of the odd numbers is identifiable. (b) Determine all subsets

S

that are identifiable.

Problem Statement